Multi-band digital compensator for a non-linear system

ABSTRACT

A pre-distorter that both accurately compensates for the non-linearities of a radio frequency transmit chain, and that imposes as few computation requirements in terms of arithmetic operations, uses a diverse set of real-valued signals that are derived from separate band signals that make up the input signal. The derived real signals are passed through configurable non-linear transformations, which may be adapted during operation, and which may be efficiently implemented using lookup tables. The outputs of the non-linear transformations serve as gain terms for a set of complex signals, which are functions of the input, and which are summed to compute the pre-distorted signal. A small set of the complex signals and derived real signals may be selected for a particular system to match the classes of non-linearities exhibited by the system, thereby providing further computational savings, and reducing complexity of adapting the pre-distortion through adapting of the non-linear transformations.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a Continuation-In-Part (CIP) of U.S. applicationSer. No. 16/408,979, filed May 10, 2019, which claims the benefit ofU.S. Provisional Application No. 62/747,994, and U.S. ProvisionalApplication No. 62/670,315, filed on May 11, 2018. This application isalso a Continuation-In-Part (CIP) of PCT Application No.PCT/US2019/031714, which claims the benefit of U.S. ProvisionalApplication No. 62/747,994, and U.S. Provisional Application No.62/670,315, filed on May 11, 2018. This application also claims thebenefit of U.S. Provisional Application No. 62/804,986, filed on Feb.13, 2019. The above-referenced applications are incorporated herein byreference.

BACKGROUND

This invention relates to digital compensation of a non-linear circuitor system, for instance linearizing a non-linear power amplifier andradio transmitter chain with a multi-band input, and in particular toeffective parameterization of a digital pre-distorter used for digitalcompensation.

One method for compensation of such a non-linear circuit is to“pre-distort” (or “pre-invert”) the input. For example, an ideal circuitoutputs a desired signal u[.] unchanged (or purely scaled or modulated),such that y[.]=u[.], while the actual non-linear circuit has aninput-output transformation y[.]=F(u[.]), where the notation y[.]denotes a discrete time signal. A compensation component is introducedbefore the non-linear circuit that transforms the input u[.], whichrepresents the desired output, to a predistorted input v[.] according toa transformation v[.]=C(u[.]). Then this predistorted input is passedthrough the non-linear circuit, yielding y[.]=F(v[.]). The functionalform and selectable parameters values that specify the transformation C() are chosen such that y[.]≈u[.] as closely as possible in a particularsense (e.g., minimizing mean squared error), thereby linearizing theoperation of tandem arrangement of the pre-distorter and the non-linearcircuit as well as possible.

In some examples, the DPD performs the transformation of the desiredsignal u[.] to the input y[.] by using delay elements to form a set ofdelayed versions of the desired signal (up to a maximum delay τ_(P)),and then using a non-linear polynomial function of those delayed inputs.In some examples, the non-linear function is a Volterra series:

y[n]=x ₀+Σ_(p)Σ_(τ) ₁ _(, . . . ,τ) _(p) x _(p)(τ₁, . . .,τ_(P))Π_(j=1 . . . p) u[n−τ _(j)]

or

y[n]=x ₀+Σ_(p)Σ_(τ) ₁ _(, . . . ,τ) _(2p−1) x _(p)(τ₁, . . .,τ_(p))Π_(j=1 . . . p) u[n−τ _(j)]Π_(j=p+1 . . . 2p−1) u[n−τ _(j)]*

In some examples, the non-linear function uses a reduced set of Volterraterms or a delay polynomial:

y[n]=x ₀+Σ_(p)Σ_(τ) x _(p)(τ)u[n−τ]|u[n−τ| ^((p−1)).

In these cases, the particular compensation function C is determined bythe values of the numerical configuration parameters x_(p).

In the case of a radio transmitter, the desired input u[.] may be acomplex discrete time baseband signal of a transmit band, and y[.] mayrepresent that transmit band as modulated to the carrier frequency ofthe radio transmitter by the function F( ) that represents the radiotransmit chain. That is, the radio transmitter may modulate and amplifythe input v[.] to a (real continuous-time) radio frequency signal p(.)which when demodulated back to baseband, limited to the transmit bandand sampled, is represented by y[.].

There is a need for a pre-distorter with a form that both accuratelycompensates for the non-linearities of the transmit chain, and thatimposes as few computation requirements in terms of arithmeticoperations to be performed to pre-distort a signal and in terms of thestorage requirements of values of the configuration parameters. There isalso a need for the form of the pre-distorter to be robust to variationin the parameter values and/or to variation of the characteristics ofthe transmit chain so that performance degradation of pre-distortiondoes not exceed that which may be commensurate with the degree of suchvariation.

In some systems, the input to a radio transmit chain is made up ofseparate channels occupying distinct frequency bands, generally withfrequency regions separating those bands in which no transmission isdesired. In such a situation, linearization of the circuit (e.g., thepower amplifier) has the dual purpose of improving the linearity of thesystem in search of the distinct frequency bands, and reducing unwantedemissions between the bands. For example, interaction between the bandsresulting from intermodulation distortion may cause such unwantedemission.

One approach to linearizing a system with a multi-band input isessentially to ignore the multi-band nature of the input. However, suchan approach may require substantial computation resources, and requirerepresentation of the input signal and predistorted signal at a highsampling rate in order to capture the non-linear interactions betweenbands. Another approach is to linearize each band independently.However, ignoring the interaction between bands generally yields poorresults. Some approaches have relaxed the independent linearization ofeach band by adapting coefficients of non-linear functions (e.g.,polynomials) based on more than one band. However, there remains a needfor improved multi-band linearization and/or reduced computationassociated with such linearization.

SUMMARY

In one aspect, in general, a pre-distorter that both accuratelycompensates for the non-linearities of a radio frequency transmit chain,and that imposes as few computation requirements in terms of arithmeticoperations and storage requirements, uses a diverse set of real-valuedsignals that are derived from the input signal, for example fromseparate band signals and their combinations, as well as optional inputenvelope and other relevant measurements of the system. The derived realsignals are passed through configurable non-linear transformations,which may be adapted during operation based on sensed output of thetransmit chain, and which may be efficiently implemented using lookuptables. The outputs of the non-linear transformations serve as gainterms for a set of complex signals, which are transformations of theinput or transformations of separate bands or combinations of separatebands of the input. The gain-adjusted complex signals are summed tocompute the pre-distorted signal, which is passed to the transmit chain.A small set of the complex signals and derived real signals may beselected for a particular system to match the non-linearities exhibitedby the system, thereby providing further computational savings, andreducing complexity of adapting the pre-distortion through adapting ofthe non-linear transformations.

In another aspect, in general, a method of signal predistortionlinearizes a non-linear circuit. An input signal (u) is processed toproduce multiple transformed signals (w). The transformed signals areprocessed to produce multiple phase-invariant derived signals (r). Thesephase-invariant derived signals (r) are determined such that eachderived signal (r_(j)) is equal to a non-linear function of one or moreof the transformed signals. The derived signals are phase-invariant inthe sense that a change in the phase of a transformed signal does notchange the value of the derive signal. At least some of the derivedsignals are equal to functions of different one or more of thetransformed signals. A distortion term is then formed by accumulatingmultiple terms.

Each term is a product of a transformed signal of the transformedsignals and a time-varying gain. The time-varying gain is a function (Φ)of one or more of the phase-invariant derived signals. The function ofthe one or more of the phase-invariant derived signals is decomposableinto a combination of one or more parametric functions (ϕ) of acorresponding single one of the phase invariant derived signals (r_(j))yielding a corresponding one of the time-varying gain components(g_(i)). An output signal (v) is determined from the distortion term andprovided for application to the non-linear circuit.

In another aspect, in general, a method of signal predistortion forlinearizing a non-linear circuit involves processing an input signal (u)that comprises multiple separate band signals (u₁, . . . , u_(N) _(b) ),where each separate band signal has a separate frequency range withinthe input frequency range of the input signal and at least part of theinput frequency range contains none of the separate frequency ranges.The processing produces a set of transformed signals (w), thetransformed signals including at least one transformed signal equal to acombination of multiple separate band signals. Multiple phase-invariantderived signals (r) are determined to be equal to respective non-linearfunctions of one or more of the transformed signals. The phase-invariantderived signals (r) are transformed according to a multiple parametricnon-linear transformations (Φ) to produce a set of gain components (g).A distortion term is formed by accumulating multiple terms (indexed byk), with each term being a combination of a transformed signal (w_(a)_(k) ) of the transformed signals and respective one or moretime-varying gain components (g_(i), i∈Λ_(k)) of the set of gaincomponents. An output signal (v) determined from the distortion term isprovided for application to the non-linear circuit.

Aspects may include one or more of the following features.

The non-linear circuit includes a radio-frequency section including aradio-frequency modulator configured to modulate the output signal to acarrier frequency to form a modulated signal and an amplifier foramplifying the modulated signal.

The input signal (u) includes quadrature components of a baseband signalfor transmission via the radio-frequency section. For example, the inputsignal (u) and the transformed signals (w) comprise complex-valuedsignals with the real and imaginary parts of the complex signalrepresenting the quadrature components.

The input signal (u) and the transformed signals (w) are complex-valuedsignals.

Processing the input signal (u) to produce the transformed signals (w)includes forming at least one of the transformed signals as a linearcombination of the input signal (u) and one or more delayed versions ofthe input signal.

At least one of the transformed signals is formed as a linearcombination includes forming a linear combination with at least oneimaginary or complex multiple input signal or a delayed version of theinput signal.

Forming at least one of the transformed signals, w_(k) to be a multipleof D_(α)w_(a)+j^(d)w_(b), where w_(a) and w_(b) are other of thetransformed signals, and D_(α) represents a delay by α, and d is aninteger between 0 and 3.

Forming the at least one of the transformed signals includes timefiltering the input signal to form said transformed signal. The timefiltering of the input signal includes applying afinite-impulse-response (FIR) filter to the input signal, or applying aninfinite-impulse-response (IIR) filter to the input signal.

The transformed signals (w) include non-linear functions of the inputsignal (u).

The non-linear functions of the input signal (u) include at least onefunction of a form u[n−τ]|u[n−τ]|^(p) for a delay τ and an integer powerp or Π_(j=1 . . . p)u[n−τ_(j)]Π_(j=p+1 . . . 2p−1)u[n−τ_(j)]* for a setfor integer delays τ₁ to τ_(2p−1), where indicates a complex conjugateoperation.

Determining a plurality of phase-invariant derived signals (r) comprisesdetermining real-valued derived signals.

Determining the phase-invariant derived signals (r) comprises processingthe transformed signals (w) to produce a plurality of phase-invariantderived signals (r).

Each of the derived signals is equal to a function of one of thetransformed signals.

Processing the transformed signals (w) to produce the phase-invariantderived signals includes, for at least one derived signal (r_(p)),computing said derived signal by first computing a phase-invariantnon-linear function of one of the transformed signals (w_(k)) to producea first derived signal, and then computing a linear combination of thefirst derived signal and delayed versions of the first derived signal todetermine at least one derived signal.

Computing a phase-invariant non-linear function of one of thetransformed signals (w_(k)) comprises computing a power of a magnitudeof the one of the transformed signals (I wk IP) for an integer powerp≥1. For example, p=1 or p=2.

Computing the linear combination of the first derived signal and delayedversions of the first derived signal comprises time filtering the firstderived signal. Time filtering the first derived signal can includeapplying a finite-impulse-response (FIR) filter to the first derivedsignal or applying an infinite-impulse-response (IIR) filter to thefirst derived signal.

Processing the transformed signals (w) to produce the phase-invariantderived signals includes computing a first signal as a phase-invariantnon-linear function of a first signal of the transformed signals, andcomputing a second signal as a phase-invariant non-linear function of asecond of the transformed signals, and then computing a combination ofthe first signal and the second signal to form at least one of thephase-invariant derived signals.

At least one of the phase-invariant derived signals is equal to afunction for two of the transformed signals w_(a) and w_(b) with a form|w_(a)[t]|^(α)|w_(b)[t−τ]|^(β) for positive integer powers a and β.

The transformed signals (w) are processed to produce the phase-invariantderived signals by computing a derived signal r_(k)[t] using at leastone of the following transformations:

r_(k)[t]=|w_(a)[t]|^(α), where α>0 for a transformed signal w_(a)[t];r_(k)[t]=0.5(1−θ+r[t−α]+θr_(b)[t]), where θ∈({1,−1}, a,b∈{1, . . . ,k−1}, and a is an integer, and r_(a)[t] and r_(b)[t] are other of thederived signals;r_(k)[t]=r_(a)[t−α]r_(b)[t], where a,b∈{1, . . . , k−1} and α is aninteger and r_(a)[t] and r_(b)[t] are other of the derived signals; andr_(k)[t]=r_(k)[t−1]+2^(−d)(r_(a)[t]−r_(k)[t−1]), where a∈{1, . . . ,k−1} and d is an integer d>0.

The time-varying gain components comprise complex-valued gaincomponents.

The method includes transforming a first derived signal (r_(j)) of theplurality of phase-invariant derived signals according to one or moredifferent parametric non-linear transformation to produce acorresponding time-varying gain components.

The one or more different parametric non-linear transformationscomprises multiple different non-linear transformations producingcorresponding time-varying gain components.

Each of the corresponding time-varying gain components forms a part of adifferent term of the plurality of terms of the sum forming thedistortion term.

Forming the distortion term comprises forming a first sum of products,each term in the first sum being a product of a delayed version of thetransformed signal and a second sum of a corresponding subset of thegain components.

The distortion term δ[t] has a form

${\delta \lbrack t\rbrack} = {\sum\limits_{k}{{w_{a_{k}}\left\lbrack {t - d_{k}} \right\rbrack}{\sum\limits_{i \in \Lambda_{k}}{g_{i}\lbrack t\rbrack}}}}$

wherein for each term indexed by k, a_(k) selects the transformedsignal, d_(k) determines the delay of said transformed signal, and Λ_(k)determines the subset of the gain components.

Transforming a first derived signal of the derived signals according toa parametric non-linear transformation comprises performing a tablelookup in a data table corresponding to said transformation according tothe first derived signal to determine a result of the transforming.

The parametric non-linear transformation comprises a plurality ofsegments, each segment corresponding to a different range of values ofthe first derived signal, and wherein transforming the first derivedsignal according to the parametric non-linear transformation comprisesdetermining a segment of the parametric non-linear transformation fromthe first derived signal and accessing data from the data tablecorresponding to a said segment.

The parametric non-linear transformation comprises a piecewise linear ora piecewise constant transformation, and the data from the data tablecorresponding to the segment characterizes endpoints of said segment.

The non-linear transformation comprises a piecewise lineartransformation, and transforming the first derived signal comprisesinterpolating a value on a linear segment of said transformation.

The method further includes adapting configuration parameters of theparametric non-linear transformation according to sensed output of thenon-linear circuit.

The method further includes acquiring a sensing signal (y) dependent onan output of the non-linear circuit, and wherein adapting theconfiguration parameters includes adjusting said parameters according toa relationship of the sensing signal (y) and at least one of the inputsignal (u) and the output signal (v).

Adjusting said parameters includes reducing a mean squared value of asignal computed from the sensing signal (y) and at least one of theinput signal (u) and the output signal (v) according to said parameters.

Reducing the mean squared value includes applying a stochastic gradientprocedure to incrementally update the configuration parameters.

Reducing the mean squared value includes processing a time interval ofthe sensing signal (y) and a corresponding time interval of at least oneof the input signal (u) and the output signal (v).

The method includes performing a matrix inverse of a Gramian matrixdetermined from the time interval of the sensing signal and acorresponding time interval of at least one of the input signal (u) andthe output signal (v).

The method includes forming the Gramian matrix as a time averageGramian.

The method includes performing coordinate descent procedure based on thetime interval of the sensing signal and a corresponding time interval ofat least one of the input signal (u) and the output signal (v).

Transforming a first derived signal of the plurality of derived signalsaccording to a parametric non-linear transformation comprises performinga table lookup in a data table corresponding to said transformationaccording to the first derived signal to determine a result of thetransforming, and wherein adapting the configuration parameterscomprises updating values in the data table.

The parametric non-linear transformation comprises a greater number ofpiecewise linear segments than adjustable parameters characterizing saidtransformation.

The non-linear transformation represents a function that is a sum ofscaled kernels, a magnitude scaling each kernel being determined by adifferent one of the adjustable parameters characterizing saidtransformation.

Each kernel comprises a piecewise linear function.

Each kernel is zero for at least some range of values of the derivedsignal.

The parametric non-linear transformations are adapted according tomeasured characteristics of the non-linear circuit.

The transformed signals include a degree-1 combination of the separateband signals.

The transformed signals include a degree-2 or a degree-0 combination ofthe separate band signals.

Each derived signal (r_(j)) of the derived signals is equal to anon-linear function of a respective subset of one or more of thetransformed signals, and at least some of the derived signals are equalto functions of different one or more of the transformed signals.

One or more of the derived signal (r_(j)) of the phase-invariant derivedsignals are transformed according to respective one or more parametricnon-linear transformations (ϕ_(i,j)) to produce a time-varying gaincomponent (g_(i)) of a plurality of gain components (g).

Each of the parametric non-linear transformations (Φ) is decomposableinto a combination of one or more parametric functions (ϕ) of acorresponding single one of the derived signals (r_(j)).

The input signal (u) is filtered (e.g., time domain filtered) to formthe plurality of separated band signals (u₁, . . . , u_(N) _(b) ).Alternatively, the separate band signals are directly provided as inputrather than the overall input signal (u).

Each of the separated band signals is represented at a same samplingrate as the input signal.

The processing of the input signal (u) to produce a plurality oftransformed signals (w) includes forming at least some of thetransformed signals as combinations of subsets of the separate bandsignals or signals derived from said separate band signals.

The combinations of subsets of the separate band signals or signalsderived from said separate band signals make use of delay,multiplication, and complex conjugate operations on the separate bandsignals.

Processing the input signal (u) to produce the plurality of transformedsignals (w) includes scaling a magnitude of a separate band signalaccording to an overall power of the input signal (r₀).

Processing the input signal (u) to produce the plurality of transformedsignals (w) includes raising a magnitude of a separate band signal to afirst exponent (α) and rotating a phase of said band signal according toa second exponent (β) not equal to the first exponent.

Processing the input signal (u) to produce the plurality of transformedsignals (w) includes forming at least one of the transformed signals asa multiplicative combination of one of the separate band signals (u_(a))and a delayed version of another of the separate band signals (u_(b)).

Forming at least one of the transformed signals as a linear combinationincludes forming a linear combination with at least one imaginary orcomplex multiple input signal or a delayed version of the input signal.

At least one of the transformed signals, w_(k), is formed to be amultiple of D_(α)w_(a)+j^(d)w_(b), where w_(a) and w_(b) are other ofthe transformed signals each of which depend on only a single one of theseparate band signals, and D_(α) represents a delay by α, and d is aninteger between 0 and 3.

In another aspect, in general, a digital predistorter circuit isconfigured to perform all the steps of any of the methods set forthabove.

In another aspect, in general, a design structure is encoded on anon-transitory machine-readable medium. The design structure compriseselements that, when processed in a computer-aided design system,generate a machine-executable representation of the digitalpredistortion circuit that is configured to perform all the steps of anyof the methods set forth above.

In another aspect, in general, a non-transitory computer readable mediais programmed with a set of computer instructions executable on aprocessor. When these instructions are executed, they cause operationsincluding all the steps of any of the methods set forth above.

DESCRIPTION OF DRAWINGS

FIG. 1 is a block diagram of a radio transmitter.

FIG. 2 is a block diagram of the pre-distorter of FIG. 1.

FIG. 3 is a block diagram of a distortion signal combiner of FIG. 2.

FIGS. 4A-E are graphs of example gain functions.

FIG. 5 is a diagram of a table-lookup implementation of a gain lookupsection of FIG. 2.

FIG. 6A-B are diagrams of a section of a table lookup for piecewiselinear functions.

FIG. 7A is a frequency plot of a two-band example with high-orderintermodulation distortion terms.

FIG. 7B is a frequency plot of an input signal corresponding to FIG. 7A.

FIG. 7C is a frequency plot of a distortion signal corresponding to FIG.7B.

FIG. 8 is a plot of a sampled carrier signal.

DESCRIPTION

Referring to FIG. 1, in an exemplary structure of a radio transmitter100, a desired baseband input signal u[.] passes to a baseband section110, producing a predistorted signal v[.]. In the description below,unless otherwise indicated, signals such as u[.] and v[.] are describedas complex-valued signals, with the real and imaginary parts of thesignals representing the in-phase and quadrature terms (i.e., thequadrature components) of the signal. The predistorted signal v[.] thenpasses through a radio frequency (RF) section 140 to produce an RFsignal p(.), which then drives a transmit antenna 150. In this example,the output signal is monitored (e.g., continuously or from time to time)via a coupler 152, which drives an adaptation section 160. Theadaptation section also receives the input to the RF section, v[.]. Theadaptation section 150 determined values of parameters x, which arepassed to the baseband section 110, and which affect the transformationfrom u[.] to v[.] implemented by that section.

The structure of the radio transmitter 100 shown in FIG. 1 includes anoptional envelope tracking aspect, which is used to control the power(e.g., the voltage) supplied to a power amplifier of the RF section 140,such that less power is provided when the input u[.] has smallermagnitude over a short term and more power is provided when it haslarger magnitude. When such an aspect is included, an envelope signale[.] is provided from the baseband section 110 to the RF section 140,and may also be provided to the adaptation section 160.

The baseband section 110 has a predistorter 130, which implements thetransformation from the baseband input u[.] to the input v[.] to the RFsection 140. This predistorter is configured with the values of theconfiguration parameters x provided by the adaptation section 160 ifsuch adaptation is provided. Alternatively, the parameter values are setwhen the transmitter is initially tested, or may be selected based onoperating conditions, for example, as generally described in U.S. Pat.No. 9,590,668, “Digital Compensator.”

In examples that include an envelope-tracking aspect, the basebandsection 110 includes an envelope tracker 120, which generates theenvelope signal e[.]. For example, this signal tracks the magnitude ofthe input baseband signal, possibly filtered in the time domain tosmooth the envelope. In particular, the values of the envelope signalmay be in the range [0,1], representing the fraction of a full range. Insome examples, there are N_(E) such components of the signal (i.e.,e[.]=(e₁[ ], . . . , e_(N) _(E) [.]), for example, with e₁[.] may be aconventional envelope signal, and the other components may be othersignals, such as environmental measurements, clock measurements (e.g.,the time since the last “on” switch, such as a ramp signal synchronizedwith time-division-multiplex (TDM) intervals), or other user monitoringsignals. This envelope signal is optionally provided to the predistorter130. Because the envelope signal may be provided to the RF section,thereby controlling power provided to a power amplifier, and because thepower provided may change the non-linear characteristics of the RFsection, in at least some examples, the transformation implemented bythe predistorter depends on the envelope signal.

Turning to the RF section 140, the predistorted baseband signal v[.]passes through an RF signal generator 142, which modulates the signal tothe target radio frequency band at a center frequency f_(c). This radiofrequency signal passes through a power amplifier (PA) 148 to producethe antenna driving signal p(.). In the illustrated example, the poweramplifier is powered at a supply voltage determined by an envelopeconditioner 122, which receives the envelope signal e[.] and outputs atime-varying supply voltage V_(c) to the power amplifier.

As introduced above, the predistorter 130 is configured with a set offixed parameters z, and values of a set of adaptation parameters x,which in the illustrated embodiment are determined by the adaptationsection 160. Very generally, the fixed parameters determine the familyof compensation functions that may be implemented by the predistorter,and the adaptation parameters determine the particular function that isused. The adaptation section 160 receives a sensing of the signalpassing between the power amplifier 148 and the antenna 150, forexample, with a signal sensor 152 preferably near the antenna (i.e.,after the RF signal path between the power amplifier and the antenna, inorder to capture non-linear characteristics of the passive signal path).RF sensor circuitry 164 demodulates the sensed signal to produce arepresentation of the signal band y[.], which is passed to an adapter162. The adapter 162 essentially uses the inputs to the RF section,namely v[.] and/or the input to the predistorter u[.](e.g., according tothe adaptation approach implemented) and optionally e[.], and therepresentation of sensed output of the RF section, namely y[.]. In theanalysis below, the RF section is treated as implementing a generallynon-linear transformation represented as y[.]=F(v[.],e[.]) in thebaseband domain, with a sampling rate sufficiently large to capture notonly the bandwidth of the original signal u[.] but also a somewhatextended bandwidth to include significant non-linear components that mayhave frequencies outside the desired transmission band. In laterdiscussions below, the sampling rate of the discrete time signals in thebaseband section 110 is denoted as f_(s).

In the adapter 162 is illustrated in FIG. 1 and described below asessentially receiving u[t] and/or v[t] synchronized with y[t]. However,there is a delay in the signal path from the input to the RF section 140to the output of the RF sensor 164. Therefore, a synchronization section(not illustrated) may be used to account for the delay, and optionallyto adapt to changes in the delay. For example, the signals are upsampledand correlated, thereby yielding a fractional sample delay compensation,which may be applied to one or the other signal before processing in theadaptation section. Another example of a synchronizer is described inU.S. Pat. No. 10,141,961, which is incorporated herein by reference.

Although various structures for the transformation implemented by thepredistorter 130 may be used, in one or more embodiments describedbelow, the functional form implemented is

v[.]=u[.]+[.]

where

δ[.]=Δ(u[.],e[.]),

and Δ(,), which may be referred to as the distortion term, iseffectively parameterized by the parameters x. Rather than using a setof terms as outlined above for the Volterra or delay polynomialapproaches, the present approach makes use of a multiple stage approachin which a diverse set of targeted distortion terms are combined in amanner that satisfies the requirements of low computation requirement,low storage requirement, and robustness, while achieving a high degreeof linearization.

Very generally, structure of the function Δ(,) is motivated byapplication of the Kolmogorov Superposition Theorem (KST). One statementof KST is that a non-linear function of d arguments x₁, . . . ,x_(d)∈[0,1]^(d) may be expressed as

$\sum\limits_{i = 1}^{{2d} + 1}{g_{i}\left( {\sum\limits_{j = 1}^{d}{h_{ij}\left( x_{j} \right)}} \right)}$

for some functions g_(i) and h_(ij). Proofs of the existence of suchfunctions may concentrate on particular types of non-linear functions,for example, fixing the h_(ij) and proving the existence of suitableg_(i). In application to approaches described in this document, thismotivation yields a class of non-linear functions defined by constituentnon-linear functions somewhat analogous to the g_(i) and/or the h_(ij)in the KST formulation above.

Referring to FIG. 2, the predistorter 130 performs a series oftransformations that generate a diverse set of building blocks forforming the distortion term using an efficient table-driven combination.As a first transformation, the predistorter includes a complextransformation component 210, labelled L_(C) and also referred to as the“complex layer.” Generally, the complex layer receives the input signal,and outputs multiple transformed signals. In the present embodiment, theinput to the complex transformation component is the complex inputbaseband signal, u[.], and the output is a set of complex basebandsignals, w[.], which may be represented as a vector of signals andindexed w₁[.], w₂[.], . . . , w_(N) _(W) [.], where N_(W) is the numberof such signals. Very generally, these complex baseband signals formterms for constructing the distortion term. More specifically, thedistortion term is constructed as a weighted summation of the set ofbaseband signals, where the weighting is time varying, and determinedbased on both the inputs to the predistorter 130, u[.] and e[.], as wellas the values of the configuration parameters, x. Going forward, thedenotation of signals with “[.]” is omitted, and the context should makeevident when the signal as a whole is referenced versus a particularsample.

Note that as illustrated in FIG. 2, the complex layer 210 is configuredwith values of fixed parameters z, but does not depend of the adaptationparameters x. For example, the fixed parameters are chosen according tothe type of RF section 140 being linearized, and the fixed parametersdetermine the number N_(W) of the complex signals generated, and theirdefinition.

In one implementation, the set of complex baseband signals includes theinput itself, w₁=u, as well as well as various delays of that signal,for example, w_(k)=u[t−k+1] for k=1, . . . , N_(W). In anotherimplementation, the complex signals output from the complex layer arearithmetic functions of the input, for example

(u[t]+u[t−1])/2;

(u[t]+ju[t−1])/2; and

((u[t]+u[t−1])/2+u[t−2])/2.

In at least some examples, these arithmetic functions are selected tolimit the needed computational resources by having primarily additiveoperations and multiplicative operations by constants that may beimplemented efficiently (e.g., division by 2). In anotherimplementation, a set of relatively short finite-impulse-response (FIR)filters modify the input u[t] to yield w_(k)[t], where the coefficientsmay be selected according to time constants and resonance frequencies ofthe RF section.

In yet another implementation, the set of complex baseband signalsincludes the input itself, w₁=u, as well as well as variouscombinations, for example, of the form

w _(k)=0.5(D _(α) w _(a) +j ^(d) w _(b)),

where D_(α) represents a delay of a signal by an integer number αsamples, and d is an integer, generally with d∈{0,1, 2, 3} may depend onk, and k>a,b (i.e., each signal w_(k) may be defined in terms ofpreviously defined signals), such that

w _(k)[t]=0.5(w _(a)[t−α]+j ^(d) w _(b)[t]).

There are various ways of choosing which combinations of signals (e.g.,the a,b,d values) determine the signals constructed. One way isessentially by trial and error, for example, adding signals from a setof values in a predetermined range that most improve performance in agreedy manner (e.g., by a directed search) one by one.

Continuing to refer to FIG. 2, a second stage is a real transformationcomponent 220, labelled L_(R) and also referred to as the “real layer.”The real transformation component receives the N_(W) signals w,optionally as well as the envelope signal e, and outputs N_(R)(generally greater than N_(W)) real signals r, in a bounded range, inthis implementation in a range [0,1]. In some implementations, the realsignals are scaled, for example, based on a fixed scale factor that isbased on the expected level of the input signal u. In someimplementations, the fixed parameters for the system may include a scale(and optionally an offset) in order to achieve a typical range of [0,1].In yet other implementations, the scale factors may be adapted tomaintain the real values in the desired range.

In one implementation, each of the complex signals wk passes to one ormore corresponding non-linear functions ƒ(w), which accepts a complexvalue and outputs a real value r that does not depend on the phase ofits input (i.e., the function is phase-invariant). Examples of thesenon-linear functions, with an input u=u_(re)+ju_(im) include thefollowing:

|w|=|w _(re) +jw _(im)|=(w _(re) ² +w _(im) ²)^(1/2);

ww*=|w| ²;

log(a+ww*); and

|w| ^(1/2).

In at least some examples, the non-linear function is monotone ornon-decreasing in norm (e.g., an increase in |w| corresponds to anincrease in r=ƒ(u)).

In some implementations, the output of a non-linear, phase-invariantfunction may be filtered, for example, with a real linear time-invariantfilters. In some examples, each of these filters is an InfiniteImpulse-Response (IIR) filter implemented as having a rationalpolynomial Laplace or Z Transform (i.e., characterized by the locationsof the poles and zeros of the Transform of the transfer function). Anexample of a Z transform for an IIR filter is:

$\frac{Y(z)}{X(z)} = \frac{z - q}{z^{2} - {2{qz}} + p}$

where, for example, p=0.7105 and q=0.8018. In other examples, a FiniteImpulse-Response (FIR). An example of a FIR filter with input x andoutput y is:

${{y\left\lbrack {n + 1} \right\rbrack} = {\sum\limits_{\tau}{\left( {1 - 2^{- k}} \right)^{\tau}{x\left\lbrack {n - \tau} \right\rbrack}}}},$

for example with k=1 or k=4.

In yet another implementation, the particular signals are chosen (e.g.,by trial and error, in a directed search, iterative optimization, etc.)from one or more of the following families of signals:

a. r_(k)=e_(k) for k=1, . . . , N_(E), where e₁, . . . , e_(N) _(E) arethe optional components of signal e;b. r_(k)[t]=|w_(a)[t]|^(α) for all t, where α>0 (with α=1 or α=2 beingmost common) and a∈{1, . . . , N_(W)} may depend on k;c. r_(k)[t]=0.5(1−θ+r_(a)[t−α]+θr_(b)[t]) for all t, where θ∈{1,−1},a,b∈{1, . . . , k−1}, and α is an integer that may depend on k;d. r_(k)[t]=r_(a)[t−α]r_(b)[t] for all t, where a,b∈{1, . . . , k−1} andα is an integer that may depend on k;e. r_(k)[t]=r_(k)[t−1]+2^(−d)(r_(a)[t]−r_(k)[t−1]) for all t, whereα∈{1, . . . , k−1} and integer d, d>0, may depend on k (equivalently,r_(k) is the response of a first order linear time invariant (LTI)filter with a pole at 1−2^(−d), applied to r_(a) for some a<k;f. r_(k) is the response (appropriately scaled and centered) of a secondorder LTI filter with complex poles (carefully selected for easyimplementability), applied to r_(a) for some a∈{1, . . . , k−1}.

As illustrated in FIG. 2, the real layer 220 is configured by the fixedparameters z, which determine the number of real signals N_(R), andtheir definition. However, as with the complex layer 210, the real layerdoes not depend on the adaptation parameters x. The choice of realfunctions may depend on characteristics of the RF section 140 in ageneral sense, for example, being selected based on manufacturing ordesign-time considerations, but these functions do not generally changeduring operation of the system while the adaptation parameters x may beupdated on an ongoing basis in at least some implementations.

According to construction (a), the components of e are automaticallytreated as real signals (i.e., the components of r). Construction (b)presents a convenient way of converting complex signals to real oneswhile assuring that scaling the input u by a complex constant with unitabsolute value does not change the outcome (i.e., phase-invariance).Constructions (c) and (d) allow addition, subtraction, and (if needed)multiplication of real signals. Construction (e) allows averaging (i.e.,cheaply implemented low-pass filtering) of real signals and construction(f) offers more advanced spectral shaping, which is needed for somereal-world power amplifiers 148, which may exhibit a second orderresonance behavior. Note that more generally, the transformationsproducing the r components are phase invariant in the original basebandinput u, that is, multiplication of u[t] by exp(jθ) or exp(jωt) does notchange r_(p)[t].

Constructing the signals w and r can provide a diversity of signals fromwhich the distortion term may be formed using a parameterizedtransformation. In some implementations, the form of the transformationis as follows:

${\delta \lbrack t\rbrack} = {\sum\limits_{k}{{w_{a_{k}}\left\lbrack {t - d_{k}} \right\rbrack}{{\Phi_{k}^{(x)}\left( {r\lbrack t\rbrack} \right)}.}}}$

The function Φ_(k) ^((x))(r) takes as an argument the N_(R) componentsof r, and maps those values to a complex number according to theparameters values of x. That is, each function Φ_(k) ^((x))(r)essentially provides a time-varying complex gain for the k^(th) term inthe summation forming the distortion term. With up to D delays (i.e.,0≤d_(k),D) and N_(W) different w[t] functions, there are up to N_(W)Dterms in the sum. The selection of the particular terms (i.e., thevalues of a_(k) and d_(k)) is represented in the fixed parameters z thatconfigure the system.

Rather than configuring functions of N_(R) arguments, some embodimentsstructure the Φ_(k) ^((x))(r) functions as a summation of functions ofsingle arguments as follows:

${\Phi_{k}^{(x)}\left( {r\lbrack t\rbrack} \right)} = {\sum\limits_{j}{\varphi_{k,j}\left( {r_{j}\lbrack t\rbrack} \right)}}$

where the summation over j may include all N_(R) terms, or may omitcertain terms. Overall, the distortion term is therefore computed toresult in the following:

${\delta \lbrack t\rbrack} = {\sum\limits_{k}{{w_{a_{k}}\left\lbrack {t - d_{k}} \right\rbrack}{\sum\limits_{j}{{\varphi_{k,j}\left( {r_{j}\lbrack t\rbrack} \right)}.}}}}$

Again, the summation over j may omit certain terms, for example, aschosen by the designer according to their know-how and other experienceor experimental measurements. This transformation is implemented by thecombination stage 230, labelled L_(R) in FIG. 2. Each term in the sumover k uses a different combination of a selection of a component a_(k)of w and a delay d_(k) for that component. The sum over j yields acomplex multiplier for that combination, essentially functioning as atime-varying gain for that combination.

As an example of one term in summation that yields the distortion term,consider w₁=u, and r=|u|² (i.e., applying transformation (b) with a=1,and α=2), which together yield a term of the form uϕ(|u|²) where ϕ( ) isone of the parameterized scalar functions. Note the contrast of such aterm as compared to a simple scalar weighting of a terms u|u|², whichlack the larger number of degrees of freedom obtainable though theparameterization of ϕ( ).

Each function ϕ_(k,j)(r_(j)) implements a parameterized mapping from thereal argument r_(j), which is in the range [0,1], to a complex number,optionally limited to complex numbers with magnitudes less than or equalto one. These functions are essentially parameterized by the parametersx, which are determined by the adaptation section 160 (see FIG. 1). Inprincipal, if there are N_(W) components of w, and delays from 0 to D−1are permitted, and each component of the N_(R) components of r may beused, then there may be up to a total of N_(W)·D·N_(R) differentfunctions ϕ_(k,j)( ).

In practice, a selection of a subset of these terms are used, beingselected for instance by trial-and-error or greedy selection. In anexample of a greedy iterative selection procedure, a number of possibleterms (e.g., w and r combinations) are evaluated according to theirusefulness in reducing a measure of distortion (e.g., peak or averageRMS error, impact on EVM, etc. on a sample data set) at an iteration andone or possible more best terms are retained before proceeding to thenext iteration where further terms may be selected, with a stoppingrule, such as a maximum number of terms or a threshold on the reductionof the distortion measure. A result is that for any term k in the sum,only a subset of the N_(R) components of r are generally used. For ahighly nonlinear device, a design generally works better employing avariety of r_(k) signals. For nonlinear systems with strong memoryeffect (i.e., poor harmonic frequency response), the design tends torequire more shifts in the w_(k) signals. In an alternative selectionapproach, the best choices of w_(k) and r_(k) with given constraintsstarts with a universal compensator model which has a rich selection ofw_(k) and r_(k), and then an L1 trimming is used to restrict the terms.

Referring to FIG. 4A, one functional form for the ϕ_(k,j)(r_(j))functions, generically referred to as ϕ(r), is as a piecewise constantfunction 410. In FIG. 4A, the real part of such a piecewise constantfunction is shown in which the interval from 0.0 to 1.0 is divided into8 section (i.e., 2^(S) sections for S=3). In embodiments that use suchform, the adaptive parameters x directly represent the values of thesepiecewise constant sections 411, 412-418. In FIG. 4A, and in examplesbelow, the r axis is divided in regular intervals, in the figure inequal width intervals. The approaches described herein do notnecessarily depend on uniform intervals, and the axis may be divided inunequal intervals, with all functions using the same set of intervals ordifferent functions potentially using different intervals. In someimplementations, the intervals are determined by the fixed parameters zof the system.

Referring to FIG. 4B, another form of function is a piecewise linearfunction 420. Each section 431-438 is linear and is defined by thevalues of its endpoints. Therefore, the function 420 is defined by the 9(i.e., 2^(S)+1) endpoints. The function 420 can also be considered to bethe weighted sum of predefined kernels b_(l)(r) for l=0, . . . , L−1, inthis illustrated case with L=2^(S)+1=9. In particular, these kernels maybe defined as:

${b_{0}(r)} = \left\{ {\begin{matrix}{1 - {rL}} & {{{for}\mspace{14mu} 0} \leq r \leq {1/L}} \\0 & {otherwise}\end{matrix},{{b_{i}(r)} = \left\{ {{\begin{matrix}{1 + {\left( {r - {i/L}} \right)L}} & {{{for}\mspace{14mu} {\left( {i - 1} \right)/L}} \leq r \leq {i/L}} \\{1 - {\left( {r - {i/L}} \right)L}} & {{{{for}\mspace{14mu} {i/L}} \leq r \leq {\left( {i + 1} \right)/L}},} \\\; & {{{{for}\mspace{14mu} 0} < i < L},} \\0 & {otherwise}\end{matrix}{and}{b_{L}(r)}} = \left\{ {\begin{matrix}{1 + {\left( {r - 1} \right)L}} & {{{for}\mspace{14mu} {\left( {L - 1} \right)/L}} \leq r \leq {1/L}} \\0 & {otherwise}\end{matrix}.} \right.} \right.}} \right.$

The function 420 is then effectively defined by the weighted sum ofthese kernels as:

${f(r)} = {\sum\limits_{l = 1}^{L}{x_{l}{b_{l}(r)}}}$

where the x_(l) are the values at the endpoints of the linear segments.

Referring to FIG. 4C, different kernels may be used. For example, asmooth function 440 may be defined as the summation of weighted kernels441, 442-449. In some examples, the kernels are non-zero over arestricted range of values of r, for example, with b_(l)(r) being zerofor r outside [(i−n)/L,(i+n)/L] for n=1, or some large value of n<L.

Referring to FIG. 4D, in some examples, piecewise linear function formsan approximation of a smooth function. In the example shown in FIG. 4D,a smooth function, such as the function in FIG. 4C, is defined by 9values, the multiplier for kernel functions b₀ through b₉. This smoothfunction is then approximated by a larger number of linear sections451-466, in this case 16 section defined by 17 endpoints. 470, 471-486.As is discussed below, this results in there being 9 (complex)parameters to estimate, which are then transformed to 17 parameters forconfiguring the predistorter. Of course, different number of estimatedparameters and linear sections may be used. For example, 4 smoothkernels may be used in estimation and then 32 linear sections may beused in the runtime predistorter.

Referring to FIG. 4E, in another example, the kernel functionsthemselves are piecewise linear. In this example, 9 kernel functions, ofwhich two 491 and 492 are illustrated, are used. Because the kernelshave linear segments of length 1/16, the summation of the 9 kernelfunctions result in a function 490 that has 16 linear segments. One wayto form the kernel functions is a 1/M^(th) band interpolation filter, inthis illustration a half-band filter. In another example that is notillustrated, 5 kernels can be used to generate the 16-segment functionessentially by using quarter-band interpolation filters. The specificform of the kernels may be determined by other approaches, for example,to optimize smoothness or frequency content of the resulting functions,for example, using linear programming of finite-impulse-response filterdesign techniques.

It should also be understood that the approximation shown in FIG. 4D-Edo not have to be linear. For example, a low-order spline may be used toapproximate the smooth function, with fixed knot locations (e.g.,equally spaced along the r axis, or with knots located with unequalspacing and/or at locations determined during the adaptation process,for example, to optimize a degree of fit of the splines to the smoothfunction.

Referring to FIG. 3, the combination stage 230 is implemented in twoparts: a lookup table stage 330, and a modulation stage 340. The lookuptable stage 330, labelled L_(T), implements a mapping from the N_(R)components of r to N_(G) components of a complex vector g. Eachcomponent g_(i) corresponds to a unique function ϕ_(k,j) used in thesummation shown above. The components of g corresponding to a particularterm k have indices i in a set denoted Λ_(k). Therefore, the combinationsum may be written as follows:

${\delta \lbrack t\rbrack} = {\sum\limits_{k}{{w_{a_{k}}\left\lbrack {t - d_{k}} \right\rbrack}{\sum\limits_{i \in \Lambda_{k}}{{g_{i}\lbrack t\rbrack}.}}}}$

This summation is implemented in the modulation stage 340 shown in FIG.3. As introduced above, the values of the a_(k), d_(k), and Λ_(k) areencoded in the fixed parameters z.

Note that the parameterization of the predistorter 130 (see FIG. 1) isfocused on the specification of the functions ϕ_(k,j)( ). In a preferredembodiment, these functions are implemented in the lookup table stage330. The other parts of the predistorter, including the selection of theparticular components of w that are formed in the complex transformationcomponent 210, the particular components of r that are formed in thereal transformation component 220, and the selection of the particularfunctions ϕ_(k,j)( ) that are combined in the combination stage 230, arefixed and do not depend on the values of the adaptation parameters x.Therefore, in at least some embodiments, these fixed parts may beimplemented in fixed dedicated circuitry (i.e., “hardwired”), with onlythe parameters of the functions being adapted by writing to storagelocations of those parameters.

One efficient approach to implementing the lookup table stage 330 is torestrict each of the functions ϕ_(k,j)( ) to have a piecewise constantor piecewise linear form. Because the argument to each of thesefunctions is one of the components of r, the argument range isrestricted to [0,1], the range can be divided into 2^(S) sections, forexample, 2^(S) equal sized sections with boundaries at i2^(−S) fori∈{0,1, . . . , 2^(S)}. In the case of piecewise constant function, thefunction can be represented in a table with 2^(S) complex values, suchthat evaluating the function for a particular value of r_(j) involvesretrieving one of the values. In the case of piecewise linear functions,a table with 1+2^(S) values can represent the function, such thatevaluating the function for a particular value of r_(j) involvesretrieving two values from the table for the boundaries of the sectionthat r_(j) is within, and appropriately linearly interpolating theretrieved values.

Referring to FIG. 5, one implementation of the lookup table stage 330,in this illustration for piecewise constant functions, makes use of aset of tables (or parts of one table) 510-512. Table 510 has one row foreach function ϕ_(k,1)(r₁), table 511 has one row for each functionϕ_(k,2)(r₂), and so forth. That is, each row represents the endpoints ofthe linear segments of the piecewise linear form of the function. Insuch an arrangement, each of the tables 510-512 will in general have adifferent number of rows. Also, it should be understood that such anarrangement of separate tables is logical, and the implemented datastructures may be different, for example, with a separate array ofendpoint values for each function, not necessarily arranged in tables asshown in FIG. 5. To implement the mapping from r to g, each elementr_(j) is used to select a corresponding column in the j^(th) table, andthe values in that column are retrieved to form a portion of g. Forexample, the r₁ ^(th) column 520 is selected for the first table 410,and the values in that column are retrieved as g₁, g₂, . . . . Thisprocess is repeated for the r₂ ^(nd) column 421 of table 511, the r₃^(rd) column 522 of table 512 and so forth to determine all thecomponent values of g. In an embodiment in which piecewise linearfunctions are used, two columns may be retrieved, and the values in thecolumns are linearly interpolated to form the corresponding section ofg. It should be understood that the table structure illustrated in FIG.5 is only one example, and that other analogous data structures may beused within the general approach of using lookup tables rather thanextensive use of arithmetic functions to evaluate the functions ϕ_(k,j)(). It should be recognized that while the input r_(p) is real, theoutput g_(i) is complex. Therefore, the cells of the table can beconsidered to hold pairs of values for the real and imaginary parts ofthe output, respectively.

The lookup table approach can be applied to piecewise linear function,as illustrated in FIG. 6A for one representative transformationg_(k)=ϕ(r_(p)). The value r_(p) is first processed in a quantizer 630,which determines which segment r_(p) falls on, and output m_(p)representing that segment. The quantizer also output a “fractional” partf_(p), which represents the location of r_(p) in the interval for thatsegment. Each cell in the column 621 identified by m_(p) has twoquantities, which essentially define one endpoint and the slope of thesegment. The slope is multiplied in a multiplier 632 by the fractionalpart f_(p), and the product is added in an adder 634 to yield the valueg_(k). Of course this is only one implementation, and differentarrangements of the values stored in the table 611, or in multipletables, and the arrangement of the arithmetic operators on selectedvalues from the table to yield the value g may be used. FIG. 6B showsanother arrangement for use with piecewise linear functions. In thisarrangement, the output m_(p) selects two adjacent columns of the table,which represent the two endpoint values. Such an arrangement reduces thestorage by a factor of two as compared to the arrangement of FIG. 6A.However, because the slope of the linear segments are not stored, anadder 635 is used to take the difference between the endpoint values,and then this difference is multiplied by f_(p) and added to one of theendpoint values in the manner of FIG. 6A.

In the description above, the input u[.] is processed as a whole,without necessarily considering any multiple band structure in thesignal in computation of a distortion term δ[.] from which apredistorted output v[.]=u[.]+δ[.] is computed. In the followingdescription, we assume that there are N_(b) spectrally distinct bands,which together occupy only a part of the available bandwidth generally,and that the input can be decomposed as a sum to spectrally distinctsignals as

u[.]=u ₁[.]+u ₂[.]+ . . . +u _(N) _(b) [.].

The techniques described above may be used in combination with thefurther techniques described below targeting the multi-band nature ofthe input. That is, the multi-band techniques extend the single-bandtechniques and essentially extend them for application to multi-bandinput.

In this embodiment, the sampling rate of the input signal is maintainedin each of the band signals, such that individually each of these bandsignals are oversampled because each of the distinct bands occupies onlya fraction of the original bandwidth. However, as described below, theapproach makes use of complex combinations of these band signals, andafter such combinations a higher sampling rate is needed to representthe combinations as compared to the individual band signals. Therefore,although in alternative embodiments it is possible to down sample theband signals, and potentially represent their complex combinations atsampling rates below the sampling rate of the overall signal, thecomputational overhead and complexity of the down and up sampling doesnot warrant any reduction in underlying computation.

In one approach to processing, the multiple band input uses essentiallythe same structure as shown in FIG. 2, which is used in the single-bandcase. In particular, the complex transformation component 210, labelledL_(C) and referred to as the “complex layer,” receives the complex inputbaseband signal, u[.], and decomposes it, for example, by bandpassfiltering, into a set of band signals (u₁[.], u₂[.], . . . , u_(N) _(b)[.]) and then outputs a set of complex baseband signals, w[.], whereeach of these baseband signals is determined from a subset of one ormore of the band signals, u_(i)[.], with the output baseband signalsagain being represented as a vector of signals and indexed w₁[.], w₂[.],. . . , w_(N) _(W) [.], where N_(W) is the number of such signals.

In the multiple band case, the output signals may be computed in anumber of ways, including by applying one or more of the followingconstructions, without limitation:

-   a. w_(k)=u_(a)r₀ ^(−α) for some a∈{1, . . . , N_(b)} and α∈(0,1),    where u_(a) is the a^(th) band, and r₀=|u₁|²+ . . . +|u_(N) _(b) |²)-   b. w_(k)=w_(a)* (i.e., complex conjugate) for some k>N_(b)+1, where    the parameter a∈{1, . . . , k−1} may depend on k-   c. w_(k)=w_(a)(D_(α)w_(b)) for some k>N_(b)+1, where the integer    parameters a,b∈{1, . . . , k−1} and a may depend on k-   d. w_(k)=|w_(a)|αe^(jβ∠w) ^(a) for some k>N_(b)+1, where the integer    parameters a∈{1, . . . , k−1} and β, and the real parameter α>0 may    depend on k. This construction may be referred to as a (α,    β)-rotation function, which for α=β, reduces to a power (i.e.,    exponent) function.

Note that construction (a) depends on a single band signal u_(a)(possibly scaled by an overall power). The construction (c) mayintroduce “cross-terms”, and repeated application of that construction,along with intervening other of the constructions, can be used togenerate a wide variety of cross-terms, which may be associated withparticular distortion components. Furthermore, other constructions inaddition to or instead of those shown above may be used, includingconstructions described above for the signal-band case. For example,within-band constructions that are analogous to those used in thesingle-band case can be used, such thatw_(k)=0.5(D_(α)w_(a)+j^(d)w_(b)), with the added constraint that bothw_(a) and w_(b) depend on only a single band signal u_(i) (as isimplicitly the case in the single-band case).

Therefore, one can consider the resulting set of complex signals w_(k)as including, for each of the band signal u_(a), a subset of the w_(k)that depends only on that band signal, which can include that bandsignal unmodified, as well as processed versions of the signal includingproducts of delayed versions, complex conjugates, powers, etc. of othersignals in the subset, as well as power-scaled versions based on overallpower of the input signal. The resulting set of complex signals w_(k)then further includes a “cross-product” subset, which includes complexcombinations of two or more band signals, for example, resulting fromapplication of construction (c).

It should be recognized that for each of the separate bands, themulti-band approach described above retains the power of linearizationwithin the band, for example, based on the subset of complex signalsthat depend only on the input in that band using the structure describedabove for the single-band case. More generally, the approaches andconstructions described above for the single-band case may be combinedwith the approaches described here for the multi-band case. Themulti-and approach further adds the capability of addressing cross termsinvolving two or more bands, and effects of overall power over multipleor all of the bands. An intention of operations in the complex layer isto generate complex signals which correspond to harmonics or otherexpected distortion components that arise from the individual bandscontained in the baseband input signal u.

One way to accomplish this goal of the resulting signals havingharmonics in the baseband is to only use what are referred to herein as“degree 1” harmonics. A degree-1 term is defined as a signal that fallsat a frequency position within the baseband that is insensitive to thecarrier frequency f_(c) to which the baseband signal u is ultimatelymodulated for radio-frequency transmission. Note that, for example,construction (c) for computing the w signals of the formw_(k)=w_(a)(D_(α)w_(b)), in combination with construction (b)w_(k)=w_(a)*, can be used to yield derived signals for a form

u ₁[t]u ₁[t−1]u ₂*[t].

More specifically, the degree of a signal w_(k), which is constructed asa combination of a set of signals (e.g., from the band signals u_(i)),is defined according to rules corresponding to the construction rulespresented above: each complex signal introduced according to (a) isassigned degree 1; if w_(k) is defined via w_(a) according toconstruction (b), the degree of w_(k) is minus the degree of w_(a); ifw_(k) is defined via w_(a) and w_(b) according to construction (c), thedegree of w_(k) is the sum of degrees of w_(a) and w_(b); and if w_(k)is defined via w_(a) according to construction (d), the degree of w_(k)is the degrees of w_(a) times β.

As in the single-band case, the generated complex signals are passed tothe second stage, the real transformation component 220, labelled L_(R)and also referred to as the “real layer.” The real transformationcomponent receives the N_(W) signals w, as well as the real “envelope”signal(s) e, and outputs N_(R) (generally greater than N_(W)) realsignals r, in a bounded range, in the implementation in a range [0,1].In one implementation for the multiple band case, the particular signalsare chosen from one or more of the following families of signalsresulting for sequential (i.e., k=1, 2, . . . ) application ofconstructions selected from the following, without limitation:

-   -   a. r_(k)=e_(k) for k=1, . . . , N_(E), where e₁, . . . , e_(N)        _(E) are the components of signal e    -   b. r_(k)=Re(w_(a)w_(b)*) or r_(k)=Im(w_(a)w_(b)*), where w_(a)        and w_(b) are formed by construction (a), w_(k)=u_(a)r₀ ^(−α)        above, or are delayed versions, w_(k)=D_(α)u_(a)r₀ ^(−α) for        α≥0, of such constructions;    -   c. r_(k)=D_(α)r_(a)+θD_(β)r_(b), where θ∈{1,−1}, a,b∈{1, . . . ,        k−1}, and α, β∈        may depend on k;    -   d. r_(k)=(D_(α)r_(a))(D_(α)r_(b)) for all t, a,b∈{1, . . . ,        k−1} and a∈        may depend on k;    -   e. r_(k)[t]=r_(k)[t−1]+2^(−d) (r_(a)[t]−r_(k)[t−1]) for all t∈        , where a∈{1, . . . , k−1} and d∈        , d>0, may depend on k (equivalently, r_(k) is the response of a        first order linear time invariant (LTI) filter with a pole at        1−2^(−d), applied to r_(a) for some a<k;    -   f. r_(k) is the response (appropriately scaled and centered) of        a second order LTI filter with complex poles (carefully selected        for easy implementability)

According to construction (a), the components of e are automaticallytreated as real signals (i.e., the components of r). Construction (b)presents a convenient way of converting complex signals to real oneswhile assuring that scaling the input u by a complex constant with unitabsolute value does not change the outcome (i.e., phase-invariance).Constructions (c) and (d) allow addition, subtraction, and (if needed)multiplication of real signals. Construction (e) allows averaging ofreal signals, and construction (f) offers more advanced spectralshaping, which is needed for some PAs which show a second orderresonance behavior.

As in the single-band case, the overall distortion term is computed as asum of N_(k) terms

${\delta \lbrack t\rbrack} = {\sum\limits_{k = 1}^{N_{k}}{{w_{a_{k}}\left\lbrack {t - d_{k}} \right\rbrack}{\sum\limits_{j}{\varphi_{k,j}\left( {r_{j}\lbrack t\rbrack} \right)}}}}$

where the k^(th) term has a selected one of the complex signals indexedby a_(k) and a selected delay d_(k), and scales the complex signal w_(a)_(k) [t−d_(k)] by a sum of the estimated functions of single of the realsignals r_(j)[.]. Again, as in the single-band case, the summation overj may omit certain terms (i.e., only relying on a subset of the r_(j)),for example, as chosen by the designer according to their know-how andother experience or experimental measurements. This transformation isimplemented by the combination stage 230 in the manner described for thesingle-band case.

As introduced above, the particular constructions used to assemble thecomplex signals w_(k) and real signals r_(k) through selections of thesequences of constructions may be based on trial-and-error, analyticalprediction of impact of various terms, heuristics, and/or a search orcombinatorial optimization to select the subset for a particularsituation (e.g., for a particular power amplifier, transmission band,etc.). One possible optimization approach may make use of greedyselection of productions to add to a set of w_(k) and r_(k) signalsaccording to their impact on an overall distortion measure. In such aselection of the terms w_(k) to use in the summation of the distortionterm, these terms may be restricted to degree-1 terms.

A number of aspects of the constructions for the complex signals w_(k)are noteworthy. For example, certain cross-terms between bands (e.g.,intermodulation terms) do not scale with the power of the individualband terms. Therefore, a possible scaling of a band signal followingconstruction (a) is found to be effective, for example for α=4:

$w_{k} = {\frac{u_{i}}{\sqrt[4]{{u_{1}}^{2} + \ldots + {u_{N_{b}}}^{2}}}.}$

Note that in most single-band applications, defining real signals by an“absolute value” formula r_(i)[t]=|u_(q)[t]| may provide better resultsthan a “power” formula r_(i)[t]=|u_(q)[t]|², which may be explained, andjustified by experimental observation of the scaling properties of thenon-linear harmonics induced by typical power amplifiers (PAs): one canview i[t]=1 u_(q)[t] as the re-scaled powerr_(i)[t]=|u_(q)[t]|²/|u_(q)[t]|. However, this does not work the sameway in the multi-band case: defining r₁[t]=|u₁[t]| does not yield thebest re-scaling, as compared to the denominator depending on the totalsignal power, as in r₁[t]=|u₁[t]|²/|u[t]|, where u[t] is the totalbaseband input (i.e., the sum of all bands). To facilitate properscaling of real signals, while avoiding aliased harmonics, the originalband signals u₁; . . . ; u_(N) _(b) can pass through the re-scalingtransformation of construction (a), for example with α=4. Once therescaling has taken place, it may be more efficient to define realsignals according to construction (b), for example as

r _(k)[t]=Re{u _(q)[t]*u _(q)[t−τ]},

or

r _(k)[t]=Im{u _(q)[t]*u _(q)[t−τ]}.

Another noteworthy construction of a complex signal uses the (α, β)rotation function of construction (d). In general, in multi-band systemsfor which the carrier-frequency-to-baseband-spectral-diameter ratio issmall enough (say, less than 5), significant high order even inter-bandharmonics may be created by a power amplifier. Compensating for thoseharmonics may require performing higher-order power operations (such asu₁[t]→u₁[t]⁵) on individual band signals. In general, taking a complexnumber z to positive integer power k means multiplying its phase by k,and taking its absolute value to the k^(th) power. In predistortionapplications, the phase manipulation part of the power operation may besignificant to the overall performance, while taking the absolute valueto the power k may be counterproductive, for example, because it doesnot match with the harmonic scaling properties of common poweramplifiers and also introduces significant numerical difficulties infixed point implementations. Taking these considerations into account,use of the (α, β) rotation functions has been found effective inpractice, for example, in cancelling even harmonics.

As introduced above, restriction to degree-1 complex signals makes thepredistorter insensitive to the ultimate carrier frequency, f_(c). Moregenerally, it is not necessary to restrict w_(k) terms that are used tobe degree-1. For example, for degree 0 and degree 2 terms, the frequencylocation of the term within the baseband is not independent of thecarrier frequency. To account for this, the complex layer receives anadditional complex signal defined as

${e_{c}\lbrack n\rbrack} = {\exp \left( {{j\; 2\pi \; \frac{f_{c}}{f_{s}}n} + \varphi} \right)}$

for some preferably constant phase ϕ, where f_(c) is the carrierfrequency for RF transmission and f_(s) is the baseband samplingfrequency for the input signal u[t]. Degree 2 terms w_(k) are multipliedby e_(c) when used in the summation to determine the distortion term,and degree 0 terms are multiplied by e_(c)*.

Note that the definition of the e_(c) depends on the ratio f_(c)/f_(s)as well as the initial phase ϕ. Preferably, this signal is generatedsuch that ϕ is equal at the start (n=0) of each transmission frame sothat the parameter estimation is consistent with each parameter use.Furthermore, if the frequency ratio is irreducible, for example,f_(c)/f_(s)=7/4, then the signal e_(c) repeats every 4 samples (i.e.,e_(c)[0]=e_(c)[4]).

Referring to FIG. 7A, and example of predistortion in a two-bandsituation is illustrated with narrowband signals that are ultimatelytransmitted (i.e., as the radio frequency signal p(t)) at frequenciesf₁+f_(c) (711) and f₂+f_(c) (712), where f_(c) (701) is the RF carrierfrequency. In this example, f₁ is illustrated as negative, and f₂ isillustrated as positive. For example, f_(c)=860.16 MHz, and|f₂−f₁|=190.0 MHz. This example focusses on predistortion to addressintermodulation terms such as an 8^(th) order intermodulation term atf₁=Δf=−4f₁+4f₂ (721) and a 10^(th) order term at 2f_(c)+6f₁−4f₂ (722).Other distortion terms (723, 724) are illustrated near f₂. These termsare at frequencies −5f₁+5f₂ and 2f_(c)+5f₁−3f₂, respectively. One way toselect these terms is by identifying spectral energy at thesefrequencies, and determining the corresponding terms that might beresponsible for distortion effects at those frequencies.

In this example, the input signal u[t] is represented at a complexsampling rate f_(s)=491.52 MHz (i.e., f_(c)/f_(s)=7/4), for modulationto the range f_(c)−f_(s)/2 to f_(c)+f_(s)/2. Referring to FIG. 7B, theinput signal therefore has components u₁ (731) and u₂ (732) atfrequencies f₁ and f₂, respectively. Referring to FIG. 7C, thedistortion term δ computed as described above, therefore includes termsat frequencies −f_(c)−4f₁+4f₂ (841) and f_(c)+6f₁−4f₂ (842), for the8^(th) order and 10^(th) order terms respectively.

In this example, to address the 8^(th) order term (841), a complexsignal w_(k)=(u₁*u₂)⁴ is used. Such a term corresponds, for example, toapplication of constructions (a)-(c) above. Without compensation for thecarrier frequency, because this is a degree zero term, it would bemodulated to frequency f_(c)−4f₁+4f₂, rather than to frequency −4f₁+4f₂.Therefore as discussed above, it is multiplied by e_(c)* yielding adistortion term w_(k)=e_(c)*(u₁*u₂)⁴, which is scaled by the adaptedgain, Σ_(i∈Δ) _(k) g_(i)[t]. Similarly, the 10^(th) order term (842) maybe addressed using a complex signal w_(k)=u₁ ² (u₁u₂*)⁴, which is adegree 2 term and therefore would be multiplied by e_(c) to yield a termw_(k)=e_(c)u₁ ²(u₁u₂*)⁴ to be scaled by an adapted gain.

In scaling the 8^(th) order term w_(k)=e_(c)*(u₁*u₂)⁴, the followingreal functions may be used, without limitation:

r ₁ =|u ₁|/√{square root over (|u ₁|² +|u ₂|²)};

r ₂ =|u ₂|/√{square root over (|u ₁|² +|u ₂|²)};

r ₃ =r ₁ +r ₂;

r ₄ =r ₁ −r ₂

r ₅ =|u ₁|;

r ₆ =|u ₂|; and

r ₇ =r ₅ r ₆.

Therefore, adapted functions ϕ_(k,j)(r_(j)) for these real functions areused to compute the respective gain terms g_(i).

Referring to FIG. 8, the sampling and periodicity of e, is illustratedfor the f_(c)/f_(s)=7/4 situation shown in FIGS. 7A-C. The sampledcarrier at the sampling frequency are illustrated with the open circles,illustrating the periodicity of 4 samples.

Therefore, as described above, in both the single and multi-band cases,a configuration of a predistorter involves selection of the sequences ofconstructions used to form the complex signals w_(k) and real signalsr_(j), which are computed at runtime of the predistorter, and remainfixed for the configuration. The parameters of the nonlinear functionsϕ_(k,j)(r), each of which maps from a scalar real signal value r to acomplex value, are in general adapted during operation of the system. Asdescribed further below, these functions are constructed using piecewiselinear forms, where in general, individual parameters only or primarilyimpact a limited range of input values, in the implementation describedbelow, by scaling kernel functions that are non-zero over limited rangesof input values. A result of this parameterization is a significantdegree or robustness resulting from well-conditioned optimizations usedto determine and adapt the individual parameters for each of thenonlinear functions.

Very generally, the parameters x of the predistorter 130 (see FIG. 1),which implements the compensation function C, may be selected tominimize a distortion between a desired output (i.e., the input to thecompensator) u[.], and the sensed output of the power amplifier y[.].For example, the parameters x, which may be the values defining thepiecewise constant or piecewise linear functions ϕ, are updated, forexample, in a gradient-based iteration based on a reference pair ofsignals (u[.], y[.]), for example, adjusting the values of theparameters such that u[.]=y[.]. In some examples that make use oftables, for example with 2^(S) entries, to encode the non-linearfunctions ϕ_(k)( ), each entry may be estimated in the gradientprocedure. In other examples, a smoothness or other regularity isenforced for these functions by limiting the number of degrees offreedom to less than 2^(S), for example, by estimating the non-linearfunction as a being in the span (linear combination) of a set of smoothbasis functions. After estimating the combination of such functions, thetable is then generated.

Therefore, the adaptation section 160 essentially determines theparameters used to compute the distortion term as δ[t]=Δ(u[t−τ], . . . ,u[t−1]) in the case that τ delayed values of the input u are used. Moregenerally, τ_(d) delayed values of the input and τ_(f) look-ahead valuesof the input are used. This range of inputs is defined for notationalconveniences as q_(u)[t]=(u[t−τ_(d)], . . . , u[t+τ_(f)]). (Note thatwith the optional use of the terms e[t], these values are also includedin the q_(u)([t]) term.) This term is parameterized by values of a setof complex parameters x, therefore the function of the predistorter canbe expressed as

v[t]=C(q _(u)[t])=u[t]+Δ(q _(u)[t])

One or more approaches to determining the values of the parameter x thatdefine the function δ( ) are discussed below.

The distortion term can be viewed in a form as being a summation

${\delta \lbrack t\rbrack} = {\sum\limits_{b}{\alpha_{b}{B_{b}\left( {q_{u}\lbrack t\rbrack} \right)}}}$

where the α_(b) are complex scalars, and B_(b)( ) can be considered tobe basis functions evaluated with the argument q_(u)[t]. The quality ofthe distortion term generally relies on there being sufficient diversityin the basis functions to capture the non-linear effects that may beobserved. However, unlike some conventional approaches in which thebasis functions are fixed, and the terms α_(b) are estimated directly,or possibly are represented as functions of relatively simple argumentssuch as |u[t]|, in approaches described below, the equivalents of thebasis functions B_(b)( ) are themselves parameterized and estimatedbased on training data. Furthermore, the structure of thisparameterization provides both a great deal of diversity that permitscapturing a wide variety of non-linear effects, and efficient runtimeand estimation approaches using the structure.

As discussed above, the complex input u[t] to produce a set of complexsignals w_(k)[t] using operations such as complex conjugation andmultiplication of delayed versions of u[t] or other w_(k)[t]. Thesecomplex signals are then processed to form a set of phase-invariant realsignals r_(p)[t] using operations such as magnitude, real, or imaginaryparts, of various w_(k)[t] or arithmetic combinations of other r_(p)[t]signals. In some examples, these real values are in the range [0,1.0] or[−1.0,1.0], or in some other predetermined bounded range. The result isthat the real signals have a great deal of diversity and depend on ahistory of u[t], at least by virtue of at least some of the w_(k)[t]depending on multiple delays of u[t]. Note that computation of thew_(k)[t] and r_(p)[t] can be performed efficiently. Furthermore, variousprocedures may be used to retain only the most important of these termsfor any particular use case, thereby further increasing efficiency.

Before turning to a variety of parameter estimation approaches, recallthat the distortion term can be represented as

${\lbrack t\rbrack} = {\sum\limits_{k}{{w_{a_{k}}\left\lbrack {t - d_{k}} \right\rbrack}{{\Phi_{k}\left( {r\lbrack t\rbrack} \right)}.}}}$

where r[t] represents the entire set of the r_(p)[t] real quantities(e.g., a real vector), and Φ( ) is a parameterized complex function. Forefficiency of computation, this non-linear function is separated intoterms that each depend on a single real value as

${\Phi_{k}\left( {r\lbrack t\rbrack} \right)} = {\sum\limits_{p}{{\varphi_{k,p}\left( {r_{p}\lbrack t\rbrack} \right)}.}}$

For parameter estimation purposes, each of the scalar complex non-linearfunctions ϕ( ) may be considered to be made up of a weighted sum of thefixed real kernels b_(l)(r), discussed above with reference to FIGS.4A-D, such that

${\varphi_{k,p}\left( r_{p} \right)} = {\sum\limits_{l}{x_{k,p,l}{b_{l}\left( r_{p} \right)}}}$

Introducing the kernel form of non-linear functions into the definitionof the distortion term yields

${\delta \lbrack t\rbrack} = {\sum\limits_{k,p,l}{x_{k,p,l}{w_{a_{k}}\left\lbrack {t - d_{k}} \right\rbrack}{{b_{l}\left( {r_{p}\lbrack t\rbrack} \right)}.}}}$

In this form representing the triple (k,p,l) as b, the distortion termcan be expressed as

${{\delta \lbrack t\rbrack} = {\sum\limits_{b}{x_{b}{B_{b}\lbrack t\rbrack}}}},$

where

B _(b)[t]

B _(b)(q _(u)[t])=w _(a) _(k) [t−d _(k)]b _(l)(r _(p)[t]).

It should be recognized that for each time t, the complex valuesB_(b)[t] depends on the fixed parameters z and the input u over a rangeof times, but does not depend on the adaptation parameters x. Thereforethe complex values B_(b)[t] for all the combinations b=(k,p,l) can beused in place of the input in the adaptation procedure.

An optional approach extends the form of the distortion term tointroduce linear dependence on a set of parameter values, p₁[t], . . . ,p_(d)[t], which may, for example be obtained by monitoring temperature,power level, modulation center frequency, etc. In some cases, theenvelope signal e[t] may be introduced as a parameter. Generally, theapproach is to augment the set of non-linear functions according to aset of environmental parameters p₁[t], . . . , p_(d)[t] so thatessentially each function

ϕ_(k,p)(r)

is replaced with d linear multiples to form d+1 functions

ϕ_(k,p)(r),ϕ_(k,p)(r)p ₁[t], . . . ,ϕ_(k,p)(r)P _(d)[t].

These and other forms of interpolation of estimated functions accordingto the set of parameter values may be used, for example, with thefunctions essentially representing corner conditions that areinterpolated by the environmental parameters.

Using the extended set of (d+1) functions essentially forms the set ofbasis functions

B _(b)(q _(u)[t])

w _(a) _(k) [t−d _(k)]b _(l)(r _(j)[t])P _(d)[t]

where b represents the tuple (k,p,l,d) and p₀=1.

What should be evident is that this form achieves a high degree ofdiversity in the functions B_(b)( ), without incurring runtimecomputational cost that may be associated with conventional techniquesthat have a comparably diverse set of basis functions. Determination ofthe parameter values x_(b) generally can be implemented in one of twoaway: direct and indirect estimation. In direct estimation, the goal isto adjust the parameters x according to the minimization:

${\left. x\leftarrow{{argmin}_{x}{{{C(u)} - \left( {v - y + u} \right)}}} \right. = {{argmin}_{x}{\sum\limits_{t \in T}{{{\Delta \left( {q_{u}\lbrack t\rbrack} \right)} - \left( {{v\lbrack t\rbrack} - {y\lbrack t\rbrack}} \right)}}^{2}}}},$

where the minimization varies the function Δ(q_(u)[t]) while the termsq_(u)[t], v[t], and y[t] are fixed and known. In indirect estimation,the goal is to determine the parameters x according to the minimization

$\left. x\leftarrow{{argmin}_{x}{{{C(y)} - v}}} \right. = {{argmin}_{x}{\sum\limits_{t \in T}{{\left( {{y\lbrack t\rbrack}{\Delta \left( {q_{y}\lbrack t\rbrack} \right)}} \right) - {v\lbrack t\rbrack}}}^{2}}}$

where q_(y)[t] is defined in the same manner as q_(u)[t], except using yrather than u. Solutions to both the direct and indirect approaches aresimilar, and the indirect approach is described in detail below.

Adding a regularization term, an objective function for minimization inthe indirect adaptation case may be expressed as

${{E(x)} - {\rho {x}^{2}} + {\frac{1}{N}{\sum\limits_{t \in T}{{{e\lbrack t\rbrack} - {\sum\limits_{k}{x_{k}{B_{k}\left( {q_{y}\lbrack t\rbrack} \right)}}}}}^{2}}}},$

where e[t]=v[t]−y[t]. This can be expressed in vector/matrix form as

${E(x)} - {\rho {x}^{2}} + {\frac{1}{N}{\sum\limits_{t \in T}{{{e\lbrack t\rbrack} - {{a\lbrack t\rbrack}x}}}^{2}}}$

where

a[t]=[B _(l)(q _(y)[t]),B ₂(q _(y)[t]), . . . ,B _(n)(q _(y)[t])].

Using the form, following matrices can be computed:

${G = {\frac{1}{N}{\sum\limits_{t \in T}{{a\lbrack t\rbrack}^{\prime}{a\lbrack t\rbrack}}}}},{L = {\frac{1}{N}{\sum\limits_{t \in T}{{a\lbrack t\rbrack}^{\prime}{e\lbrack t\rbrack}}}}},{and}$$R = {\frac{1}{N}{\sum\limits_{t \in T}{{{e\lbrack t\rbrack}}^{2}.}}}$

From these, one approach to updating the parameters x is by a solution

x←(ρI _(n) +G)⁻¹ L

where I_(n) denotes an n×n identity. An alternative to performing theinversion is to use a coordinate descent approach in which at eachiteration, a single one of the parameters is updated.

In some examples, the Gramian, G, and related terms above, areaccumulated over a sampling interval T, and then the matrix inverse iscomputed. In some examples, the terms are updated in a continualdecaying average using a “memory Gramian” approach. In some suchexamples, rather than computing the inverse at each step, a coordinatedescent procedure is used in which at each iteration, only one of thecomponents of x is updated, thereby avoiding the need to perform a fullmatrix inverse, which may not be computationally feasible in someapplications.

As an alternative to the solution above, a stochastic gradient approachmay be used implementing:

x←x−ζ(a[τ]′(a[τ]x−e[τ])+ρx)

where ζ is a step size that is selected adaptively and r is a randomlyselected time sample from a buffer of past pairs (q_(y)[t], v[t])maintained, for example, by periodic updating, and random samples fromthe buffer are selected to update the parameter values using thegradient update equation above.

A modified version of the stochastic gradient approach, involvesconstructing a sequence of random variables {tilde over (x)}₀, {tildeover (x)}₁, . . . (taking values in

^(n), n-dimensional complex numbers), defined by

{tilde over (x)} _(k+1′) ={tilde over (x)} _(k)+αa[τ_(k)]′(e[τ_(k)]−a[τ_(k)]{tilde over (x)} _(k))−αρ{tilde over (x)}_(k),

where {tilde over (x)}₀=0, and τ₁, τ₂, . . . are independent randomvariables uniformly distributed over the available time buffer, and ρ>0is the regularization constant from the definition of E=E(x), and α>0 isa constant such that

α(ρ+|a[t]|²)<2

for every t. The expected value x _(k)=E[{tilde over (x)}_(k)] can beproven to converge to

x*=arg min E(x)

as k→∞. An optional additional averaging operation

{tilde over (y)} _(k+1) ={tilde over (y)} _(k)+ϵ({tilde over (x)} _(k)−{tilde over (y)} _(k))

with ϵ∈(0,1] may be used. The difference between {tilde over (y)}_(k)and x* is guaranteed to be small for large k as long as ϵ>0 is smallenough. This approach to minimizing E(x) can be referred to as a“projection” method, since the map

x

x+|a[t]|⁻² a[t]′(e[t]−a[t]x[t])

projects x onto the hyperplane defined by

a[t]x=e[t].

In practical implementations of the algorithm, the sequence of the τ_(k)is generated as a pseudo-random sequence of samples, and thecalculations of {tilde over (y)}_(k) can be eliminated (which formallycorresponds to ϵ=1, i.e., {tilde over (y)}_(k)={tilde over (x)}_(k−1)).As a rule, this requires using a value of α that results in a smallerminimal upper bound for

α(ρ+|a[t]|²)

(for example, α(ρ+|a[t]|²)<1, or α(ρ+|a[t]|²)<0.5). More generally, thevalues of α and ϵ are sometimes adjusted, depending on the progress madeby the stochastic gradient optimization process, where the progress ismeasured by comparing the average values of |e[τ_(k)]| and|e[τ_(k)]−a[τ_(k)]{tilde over (x)}_(k)|.

Another feature of a practical implementation is a regular update of theset of the optimization problem parameters a[t], e[t], as the datasamples a[t], e[t] observed in the past are being replaced by the newobservations.

Yet other adaptation procedures that may be used in conjunction with theapproaches presented in this document are described in co-pending U.S.application Ser. No. 16/004,594, titled “Linearization System,” filed onJun. 11, 2018, and published as US2019/0260401 A1 on Aug. 22, 2019,which is incorporated herein by reference.

Returning to the selection of the particular terms to be used for adevice to be linearized, which are represented in the fixed parametersz, which includes the selection of the particular w_(k) terms togenerate, and then the particular r_(p) to generate from the w_(k), andthen the particular subset of r_(p) to use to weight each of the w_(k)in the sum yielding the distortion term, uses a systematic methodology.One such methodology is performed when a new device (a “device undertest”, DUT) is evaluated for linearization. For this evaluation,recorded data sequences (u[.],y[.]) and/or (v[.],y[.]) are collected. Apredistorter structure that includes a large number of terms, possiblyan exhaustive set of terms within a constrain on delays, number of w_(k)and r_(p) terms etc. is constructed. The least mean squared (LMS)criterion discussed above is used to determine the values of theexhaustive set of parameters x. Then, a variable selection procedure isused and this set of parameters is reduced, essentially, by omittingterms that have relatively little impact on the distortion term δ[.].One way to make this selection uses the LASSO (least absolute shrinkageand selection operator) technique, which is a regression analysis methodthat performs both variable selection and regularization, to determinewhich terms to retain for use in the runtime system. In someimplementations, the runtime system is configured with the parametervalues x determined at this stage. Note that it should be understoodthat there are some uses of the techniques described above that omit theadapter completely (i.e., the adapter is a non-essential part of thesystem), and the parameters are set one (e.g., at manufacturing time),and not adapted during operation, or may be updated from time to timeusing an offline parameter estimation procedure.

An example of applying the techniques described above starts with thegeneral description of the distortion term

${\delta \lbrack t\rbrack} = {\sum\limits_{k}{{w_{a_{k}}\left\lbrack {t - d_{k}} \right\rbrack}{\sum\limits_{j}{{\varphi_{k,j}\left( {r_{j}\lbrack t\rbrack} \right)}.}}}}$

The complex signals derived from the input, and the real signals derivedfrom the complex signals are have the following full form:

${\delta \lbrack t\rbrack} = {{\sum\limits_{k = {- 5}}^{+ 5}{{u\left\lbrack {t - k} \right\rbrack}{\sum\limits_{j = {- 5}}^{+ 5}{\varphi_{1,k,j}\left( {{u\left\lbrack {t - k - j} \right\rbrack}} \right)}}}} + {\sum\limits_{l = {- 5}}^{+ 5}{\sum\limits_{d = 0}^{1}{\frac{\left( {{u\left\lbrack {t - l} \right\rbrack} + {j^{d}{u\left\lbrack {t - l - 1} \right\rbrack}}} \right)}{2}{\varphi_{2,l,d}\left( \frac{{{u\left\lbrack {t - l} \right\rbrack} + {u\left\lbrack {t - l - 1} \right\rbrack}}}{2} \right)}}}} + {\sum\limits_{m = {- 5}}^{+ 5}{\sum\limits_{n = {- 2}}^{+ 2}{{u\left\lbrack {t - m} \right\rbrack}{\varphi_{3,m,n}\left( {{{u\left\lbrack {t - m} \right\rbrack}}{{u\left\lbrack {t - m - n} \right\rbrack}}} \right)}}}}}$

This form creates a total of 198 (=121+22+55) terms. In an experimentalexample, this set of terms is reduced from 198 terms to 6 terms using aLASSO procedure. These remaining 6 terms result in the distortion termhaving the form:

${\delta \lbrack t\rbrack} = {{{u\lbrack t\rbrack}{\varphi_{1,0,0}\left( {{u\lbrack t\rbrack}} \right)}} + {{u\left\lbrack {t - 1} \right\rbrack}{\varphi_{1,1,0}\left( {{u\left\lbrack {t - 1} \right\rbrack}} \right)}} + {\frac{\left( {{u\left\lbrack {t - 4} \right\rbrack} + {{ju}\left\lbrack {t - 5} \right\rbrack}} \right)}{2}{\varphi_{2,4,1}\left( \frac{{{u\left\lbrack {t - 4} \right\rbrack} + {u\left\lbrack {t - 5} \right\rbrack}}}{2} \right)}} + {\frac{\left( {{u\left\lbrack {t + 2} \right\rbrack} + {u\left\lbrack {t + 1} \right\rbrack}} \right)}{2}{\varphi_{2,{- 2},0}\left( \frac{{{u\left\lbrack {t + 2} \right\rbrack} + {u\left\lbrack {t + 1} \right\rbrack}}}{2} \right)}} + {{u\left\lbrack {t - 5} \right\rbrack}{\varphi_{3,5,2}\left( {{{u\left\lbrack {t - 5} \right\rbrack}}{{u\left\lbrack {t - 7} \right\rbrack}}} \right)}} + {{u\left\lbrack {t + 5} \right\rbrack}{\varphi_{3,{- 5},{- 2}}\left( {{{u\left\lbrack {t + 5} \right\rbrack}}{{u\left\lbrack {t + 7} \right\rbrack}}} \right)}}}$

This form is computationally efficient because only 6 w_(k) complexsignals and 6 real signals r_(p) terms that must be computed at eachtime step. If each non-linear transformation is represented by 32 linearsegments, then the lookup tables have a total of 6 times 33, or 198complex values. If each non-linear function is represented by 32piecewise segments defined by 6 kernels, then there are only 36 complexparameter values that need to be adapted (i.e., 6 scale factors for thekernels of each non-linear function, and 6 such non-linear functions).

The techniques described above may be applied in a wide range ofradio-frequency communication systems. For example, approach illustratedin FIG. 1 may be used for wide area (e.g., cellular) base stations tolinearize transmission of one or more channels in a system adhering tostandard, such as 3GPP or IEEE standards (implemented over licensed andunlicensed frequency bands), pre-5G and 5G New Radio (NR), etc.Similarly, the approach can be implemented in a mobile station (e.g., asmartphone, handset, mobile client device (e.g., a vehicle), fixedclient device, etc.). Furthermore, the techniques are equally applicableto local area communication (e.g., “WiFi”, the family of 802.11protocols, etc.) as they are to wide area communication. Furthermore,the approaches can be applied to wired rather than wirelesscommunication, for example, to linearize transmitters in coaxial networkdistribution, for instance to linearize amplification and transmissionstages (e.g., including coaxial transmission lines) for DOCSIS (DataOver Cable Service Interface Specification) head ends system and clientmodems. For example, a real high-frequency DOCSIS signal maybe digitallydemodulated to quadrature components (e.g., a complex representation) ata lower frequency (e.g., baseband) range and the techniques describedabove may be applied to the demodulated signal. Yet other applicationsare not necessarily related to electrical signals, and the techniquesmay be used to linearize mechanical or acoustic actuators (e.g., audiospeakers), and optical transmission systems. Finally, although describedabove in the context of linearizing a transmission path, with a suitablereference signal representing a transmission (e.g. predefine pilotsignal patterns) the approach may be used to linearize a receiver, or tolinearize a combined transmitter-channel-receiver path.

A summary of a typical use case of the approaches described above is asfollows. First, initial data sequences (u[.],y[.]) and/or (v[.],y[.]),as well as corresponding sequences e[.] and p[.] in implementations thatmake use of these optional inputs, are obtained for a new type ofdevice, for example, for a new cellular base station or a smartphonehandset. Using this data, a set of complex signals w_(k) and realsignals r_(p) are selected for the runtime system, for example, based onan ad hoc selection approach, or an optimization such as using the LASSOapproach. In this selection stage, computational constraints for theruntime system are taken into account so that the computationallimitations are not exceeded and/or performance requirements are met.Such computational requirements may be expressed, for example, in termscomputational operations per second, storage requirements, and/or forhardware implementations in terms of circuit area or power requirements.Note that there may be separate limits on the computational constraintsfor the predistorter 130, which operates on every input value, and onthe adapter, which may operate only from time to time to update theparameters of the system. Having determined the terms to be used in theruntime system, a specification of that system is produced. In someimplementations, that specification includes code that will execute on aprocessor, for example, an embedded processor for the system. In someimplementations, the specification includes a design structure thatspecifies a hardware implementation of the predistorter and/or theadapter. For example, the design structure may include configurationdata for a field-programmable gate array (FPGA), or may include ahardware description language specific of an application-specificintegrated circuit (ASIC). In such hardware implementations, thehardware device includes input and output ports for the inputs andoutputs shown in FIG. 1 for the predistorter and the adapter. In someexamples, the memory for the predistorter is external to the device,while in other examples, it is integrated into the device. In someexamples, the adapter is implemented in a separate device than thepredistorter, in which case the predistorter may have a port forreceiving updated values of the adaption parameters.

In some implementations, a computer accessible non-transitory storagemedium includes instructions for causing a digital processor to executeinstructions implementing procedures described above. The digitalprocessor may be a general-purpose processor, a special purposeprocessor, such as an embedded processor or a controller, and may be aprocessor core integrated in a hardware device that implements at leastsome of the functions in dedicated circuitry (e.g., with dedicatedarithmetic units, storage registers, etc.). In some implementations, acomputer accessible non-transitory storage medium includes a databaserepresentative of a system including some or all of the components ofthe linearization system. Generally speaking, a computer accessiblestorage medium may include any non-transitory storage media accessibleby a computer during use to provide instructions and/or data to thecomputer. For example, a computer accessible storage medium may includestorage media such as magnetic or optical disks and semiconductormemories. Generally, the database (e.g., a design structure)representative of the system may be a database or other data structurewhich can be read by a program and used, directly or indirectly, tofabricate the hardware comprising the system. For example, the databasemay be a behavioral-level description or register-transfer level (RTL)description of the hardware functionality in a high-level designlanguage (HDL) such as Verilog or VHDL. The description may be read by asynthesis tool which may synthesize the description to produce a netlistcomprising a list of gates from a synthesis library. The netlistcomprises a set of gates that also represent the functionality of thehardware comprising the system. The netlist may then be placed androuted to produce a data set describing geometric shapes to be appliedto masks. The masks may then be used in various semiconductorfabrication steps to produce a semiconductor circuit or circuitscorresponding to the system. In other examples, the database may itselfbe the netlist (with or without the synthesis library) or the data set.

It is to be understood that the foregoing description is intended toillustrate and not to limit the scope of the invention, which is definedby the scope of the appended claims. Reference signs, including drawingreference numerals and/or algebraic symbols, in parentheses in theclaims should not be seen as limiting the extent of the matter protectedby the claims; their sole function is to make claims easier tounderstand by providing a connection between the features mentioned inthe claims and one or more embodiments disclosed in the Description andDrawings. Other embodiments are within the scope of the followingclaims.

What is claimed is:
 1. A method of signal predistortion for linearizinga non-linear circuit, the method comprising: processing an input signal(u) comprising a plurality of separate band signals (u₁, . . . , u_(N)_(b) ), each separate band signal having a separate frequency rangewithin the input frequency range of the input signal, at least part ofthe input frequency range containing none of the separate frequencyrange, the processing producing a plurality of transformed signals (w),the transformed signals including at least one transformed signal equalto a combination of multiple separate band signals; determining aplurality of phase-invariant derived signals (r) to be equal torespective non-linear functions of one or more of the transformedsignals; transforming the plurality of phase-invariant derived signals(r) according to a plurality of parametric non-linear transformations(Φ) to produce a plurality of gain components (g); forming a distortionterm by accumulating a plurality of terms (k), each term being acombination of a transformed signal (w_(a) _(k) ) of the plurality oftransformed signals and respective one or more time-varying gaincomponents (g_(i), i∈Λ_(k)) of the plurality of gain components; andproviding an output signal (v) determined from the distortion term forapplication to the non-linear circuit.
 2. The method of claim 1, furthercomprising adapting the plurality of parametric non-lineartransformations according to measured characteristics of the non-linearcircuit.
 3. The method of claim 1, wherein the at least one transformedsignal comprise a degree-1 combination of the separate band signals. 4.The method of claim 3, where the at least one transformed signal furthercomprise at least one degree-2 or at least one degree-0 combination ofthe separate band signals.
 5. The method of claim 1, wherein eachderived signal (r_(j)) of the plurality of derived signals is equal to anon-linear function of a respective subset of one or more of thetransformed signals, at least some of the derived signals being equal tofunctions of different one or more of the transformed signals.
 6. Themethod of claim 3, further comprising transforming one or more of thederived signal (r_(j)) of the plurality of phase-invariant derivedsignals according to respective one or more parametric non-lineartransformations (ϕ_(i,j)) to produce a time-varying gain component(g_(i)) of a plurality of gain components (g).
 7. The method of claim 1,wherein each of the parametric non-linear transformations (Φ) isdecomposable into a combination of one or more parametric functions (ϕ)of a corresponding single one of the derived signals (r_(j)).
 8. Themethod of claim 1, further comprising filtering the input signal (u) toform the plurality of separated band signals (u₁, . . . , u_(N) _(b) ).9. The method of claim 8, wherein each of the separated band signals isrepresented at a same sampling rate as the input signal.
 10. The methodof claim 1 wherein the processing of the input signal (u) to produce aplurality of transformed signals (w) includes forming at least some ofthe transformed signals as combinations of subsets of the separate bandsignals or signals derived from said separate band signals.
 11. Themethod of claim 10, wherein the combinations of subsets of the separateband signals or signals derived from said separate band signals make useof delay, multiplication, and complex conjugate operations on theseparate band signals.
 12. The method of claim 1, wherein the non-linearcircuit comprises a radio-frequency section including a radio-frequencymodulator configured to modulate the output signal to a carrierfrequency to form a modulated signal and an amplifier for amplifying themodulated signal.
 13. The method of claim 12, wherein the input signal(u) comprises quadrature components of a baseband signal fortransmission via the radio-frequency section.
 14. The method of claim 1,wherein the input signal (u) and the plurality of transformed signals(w) comprise complex-valued signals.
 15. The method of claim 1, whereinprocessing the input signal (u) to produce the plurality of transformedsignals (w) includes scaling a magnitude of a separate band signalaccording to an overall power of the input signal (r₀).
 16. The methodof claim 1, wherein processing the input signal (u) to produce theplurality of transformed signals (w) comprises raising a magnitude of aseparate band signal to a first exponent (α) and rotating a phase ofsaid band signal according to a second exponent (β) not equal to thefirst exponent.
 17. The method of claim 1, wherein processing the inputsignal (u) to produce the plurality of transformed signals (w) includesforming at least one of the transformed signals as a multiplicativecombination of one of the separate band signals (u_(a)) and a delayedversion of another of the separate band signals (u_(b)).
 18. The methodof claim 15, wherein forming at least one of the transformed signals asa linear combination includes forming a linear combination with at leastone imaginary or complex multiple input signal or a delayed version ofthe input signal.
 19. The method of claim 18, wherein forming at leastone of the transformed signals, w_(k), to be a multiple ofD_(α)w_(a)+j^(d)w_(b), where w_(a) and w_(b) are other of thetransformed signals each of which depend on only a single one of theseparate band signals, and D_(α) represents a delay by α, and d is aninteger between 0 and
 3. 20. The method of claim 15, wherein forming theat least one of the transformed signals includes time filtering theinput signal to form said transformed signal.
 21. The method of claim20, wherein time filtering the input signal includes applying afinite-impulse-response (FIR) filter to the input signal.
 22. The methodof claim 20, wherein time filtering the input signal includes applyingan infinite-impulse-response (IIR) filter to the input signal.
 23. Themethod of claim 1, wherein the plurality of transformed signals (w)includes non-linear functions of the separate band signals (u_(i)). 24.The method of claim 23, wherein the non-linear functions of the separatesignals (u_(i)) comprises at least one function of a formu _(i) |u _(j)|² ,i≠j, oru _(i) |u _(i) u _(j) |,i≠j.
 25. The method of claim 1, whereindetermining a plurality of phase-invariant derived signals (r) comprisesdetermining real-valued derived signals.
 26. The method of claim 1,wherein determining a plurality of phase-invariant derived signals (r)comprises processing the transformed signals (w) to produce a pluralityof phase-invariant derived signals (r).
 27. The method of claim 26,wherein each of the derived signals is equal to a function of one of thetransformed signals.
 28. The method of claim 26, wherein processing thetransformed signals (w) to produce a plurality of phase-invariantderived signals includes for at least one derived signal (r_(p))computing said derived signal by first computing a phase-invariantnon-linear function of one of the transformed signals (w_(k)) to producea first derived signal, and then computing a linear combination of thefirst derived signal and delayed versions of the first derived signal todetermine the at least one derived signal.
 29. The method of claim 28,wherein computing a phase-invariant non-linear function of one of thetransformed signals (w_(k)) comprises computing a power of a magnitudeof the one of the transformed signals (|w_(k)|^(p)) for an integer powerp≥1.
 30. The method of claim 29, wherein p=1 or p=2.